Intrinsic pseudodifferential calculi on any compact Lie group

In this paper, we define in an intrinsic way operators on a compact Lie group by means of symbols using the representations of the group. The main purpose is to show that these operators form a symbolic pseudo-differential calculus which coincides or generalises the (local) H\"ormander pseudo-differential calculus on the group viewed as a compact manifold.

Over the past five decades, pseudo-differential operators have become a powerful and versatile tool in the analysis of Partial Differential Equations (PDE's) in various contexts. Although they may be used for global analysis (essentially in the Euclidean setting), they can be localised and this allows one to define them on manifolds. However, on a manifold, one can no longer attach a global symbol to a single operator in the calculus (although one could recover a -partial -global definition of operators on manifolds for instance using linear connections, see [14] and the references therein). The subject of the present paper is to define globally symbolic calculi on a special class of manifolds, more precisely on any compact Lie group G. Naturally the first aim of this article is to show that the fundamental properties of the calculi hold true, thereby justifying the vocabulary. We will also show that the resulting calculi coincides with the Hörmander calculi localised on G viewed as a compact manifold -when the Hörmander calculi can be defined. These results are new, even in the case of the tori. Several applications to PDE's have already appeared in other publications: e.g. construction of parametrices, study of global hypoellipticity, see [10,11] and references therein. However the motivation for this paper lies primarily in giving a sound basis for the underlying analysis and in highlighting the methods used in the proofs as their generalisations seem tantalising close.
It is quite natural to define pseudo-differential operators globally on the torus by using Fourier series and considering symbols as functions of a variable in the torus and another variable in the integer lattice (for an overview of toroidal quantisation, especially Agranovitch's contributions, see [12]). Michael Taylor argued in his monograph [17,Section I.2] that an analogue quantisation is formally true on any Lie group of type 1, considering again symbols as functions of a variable of the group G and another variable of its dual G (which is the set of equivalence classes of the unitary irreducible representations of G). Just afterwards, Zelditch in [20] defined a (local) symbolic pseudo-differential calculus on a hyperbolic manifold with a related quantisation. Pseudo-differential calculi have also been defined on the Heisenberg group by Taylor in [17], see also [2] and [6], and in other directions by Dynin, Folland, Beals, Greiner, Howe (see [7] and the references therein). See also [4] for a global pseudo-differential calculus on homogeneous Lie groups (although it may not qualify as symbolic, being defined in terms of properties of the kernels of the operators).
It would be nearly impossible to review in this introduction the vast literature on classes of operators defined on Lie groups (especially if one has to include all the studies of spectral multipliers of sub-Laplacians). Instead, in this article, we focus on pseudo-differential operators, in the sense that the operators are not necessarily of convolution type. Then studies of pseudo-differential calculi on Lie groups form a much shorter list and the ones known to the author are amongst the calculi mentioned directly or indirectly in this introduction.
On compact Lie groups, the dual of the group G is discrete and Taylor's quantisation yields formulae formally closed to the case of the torus. A decade ago, Michael Ruzhansky and Ville Turunen started to push even further the ideas of Taylor, Zelditch and Agranovitch in the compact case (commutative and non-commutative). In their monograph [11], they consider symbols in the sense due to Michael Taylor [17], they define difference operators on G, and they propose a definition of symbol classes. These essential concepts have opened up new possibilities for global analysis on compact Lie groups. In their monograph and in their subsequent works with Jens Wirth [10,13], they also present many applications of the resulting calculi to the study of Partial Differential Equations. However one can argue that the fundamental properties of the calculi have been left to be shown. For instance, the proof that the composition of two operators remains in the calculus [11,Theorem 10.7.8] is incomplete since it is impossible with their analysis to justify the claims in the last paragraph of their proof. Another example is the continuity on Sobolev spaces, which is shown for very special cases and using the composition property.
This paper aims at proving the fundamental properties of the symbolic calculus on compact Lie groups proposed by Michael Ruzhansky, Ville Turunen and Jens Wirth.
Following the ideas in the introduction of [2], let us formalise what is meant here by the fundamental properties of a calculus: Definition 1.1. For each m ∈ R, let Ψ m be a given Fréchet space of continuous operators D(G) → D(G). We say that the space Ψ ∞ := ∪ m Ψ m form a pseudo-differential calculus when it is an algebra of operators satisfying: (1) The continuous inclusions Ψ m ⊂ Ψ m hold for any m ≤ m .
(2) Ψ ∞ is an algebra of operators. Furthermore if T 1 ∈ Ψ m 1 , T 2 ∈ Ψ m 2 , then T 1 T 2 ∈ Ψ m 1 +m 2 , and the composition is continuous as a map Ψ m 1 × Ψ m 2 → Ψ m 1 +m 2 . (3) Ψ ∞ is stable under taking the adjoint. Furthermore if T ∈ Ψ m then T * ∈ Ψ m , and taking the adjoint is continuous as a map Ψ m → Ψ m . (4) Ψ ∞ contains the differential calculus on G. More precisely, Diff m (G) ⊂ Ψ m (G) for every m ∈ N 0 . (5) Ψ ∞ is continuous on the Sobolev spaces with the loss of derivatives bounded by the order. Moreover, for any s ∈ R and T ∈ Ψ m , T L (H s ,H s−m ) is bounded by a semi-norm of T ∈ Ψ m , up to a constant of s, m and of the calculus.
The operator classes considered in this paper are defined in Section 3.3 and denoted by Ψ m ρ,δ (G, ∆), or just Ψ m ρ,δ , m ∈ R, 1 ≥ ρ ≥ δ ≥ 0, ρ = 0, δ = 1. The (localised) Hörmander class of operators defined on the group G viewed as a manifold is denoted by Ψ m ρ,δ (G, loc), m ∈ R, 1 ≥ ρ > δ ≥ 0, ρ ≥ 1 − δ. The conditions on the parameters ρ, δ for Ψ m ρ,δ (G, loc) comes from the necessary consistency when changing charts, and imply ρ > 1 2 . In this paper, we show that the classes of operators Ψ m ρ,δ (G, ∆) and Ψ m ρ,δ (G, loc) coincide when the latter can be defined. This is a new result and generalises the case (ρ, δ) = (1, 0) announced in [11,10]. This enables future applications to PDE's (even on the torus) when the calculus that intervenes have parameters e.g. (ρ, δ) = ( 1 2 , 1 2 ) for which the Hörmander classes on G are not defined. The ideas and methods used in this article come from the 'classical' harmonic analysis on Lie groups. We show that multipliers in the Laplace-Beltrami operator L are also in the calculus in a uniform way -hereby obtaining a new result with respect to the body of work of Michael Ruzhansky, Ville Turunen and Jens Wirth. For this, we use the well-known properties of the heat kernel of L [19] and methods regarding spectral multipliers [1] (it should be noted that the analysis of Michael Ruzhansky, Ville Turunen and Jens Wirth makes no use of the heat kernel). This enables us to use Littlewood-Payley decompositions with uniform estimates for the dyadic pieces. This also allows us to obtain precise estimates for the kernels of the operators (these estimates are also new).
It seems possible to generalise most of these ideas and methods to any unimodular Lie group of type-1 with polynomial growth of the volume and even to some of their quotients. The resulting calculi would certainly depend on the choice of a fixed left-invariant sub-Laplacian. An important technical problem would come from the fact that, on a compact Lie group, we choose the Laplace-Beltrami operator which is central and with a scalar group Fourier transform. This could no longer be assumed for a left-invariant sub-Laplacian. This paper is organised as follows. After the preliminaries in Section 2, we define in Section 3 the symbol and operator classes studied in this paper. The main result is stated in Section 3.4, where the organisation of the proofs is also explained. In Section 4, we present some first results. In Section 5, we analyse the kernels associated with operators in Ψ m ρ,δ . In Section 6, we prove that ∪ m Ψ m ρ,δ satisfies the algebraic properties in Definition 1.1. Eventually, in Section 7, we show the continuity on Sobolev spaces and this concludes the proof of the fundamental properties of the calculus ∪ m Ψ m ρ,δ . We also obtain a commutator characterisation which implies that the symbolic calculus coincides with the local Hörmander calculus. In A, we study the multipliers in L.
Notation: N 0 = {0, 1, 2, . . .} denotes the set of non-negative integers and N 0 = {1, 2, . . .} the set of positive integers. · , · denote the upper and lower integer parts of a real number. We also set (r) + := max(0, r) for any r ∈ R. If H 1 and H 2 are two Hilbert spaces, we denote by L (H 1 , H 2 ) the Banach space of the bounded operators from H 1 to H 2 . If H 1 = H 2 = H then we write L (H 1 , H 2 ) = L (H).

Preliminaries
In this section, we set the notation for the group and some of its natural structures, such as the convolution, its representations, the Plancherel formula, and the Laplace-Beltrami operator. This material is very well known, see e.g. [16,9,19,11].
2.1. Some notation for the structures on the group G. In this paper, G always denotes a connected compact Lie group and n is its dimension. Its Lie algebra g is the tangent space of G at the neutral element e G . It is always possible to define a left-invariant Riemannian distance on G, denoted by d(·, ·). We also denote by |x| = d(x, e G ) the Riemannian distance on the Riemann between x and the neutral element e G and by B(r) := {|x| < r} the ball about e G of radius r > 0. In this paper, R 0 denotes the maximum radius of the ball around the neutral element, i.e. B(R 0 ) = G, and 0 ∈ (0, 1) denotes the radius of a ball B( 0 ) which gives a chart around the neutral element for the exponential mapping exp G : g → G.
We may identify the Lie algebra g with the space of left-invariant vector fields. More precisely, if X ∈ g, then we denote by X andX the (respectively) left and right invariant vector fields given by: respectively, for x ∈ G and φ ∈ D(G). In this paper, D(G) denotes the Fréchet space of smooth functions on G. One easily checks We denote by Diff 1 (G) the space of smooth vector fields on G. It is a left D(G)-module generated by any basis of left-invariant vector fields or by any basis of right-invariant vector fields. More generally, for k ∈ N, Diff k (G) denotes the space of smooth differential operators of order k. Any element of Diff k (G) may be written as a linear combination of a α (x)X α , |α| = k, where a α ∈ D(G), and X α := X α 1 1 . . . X αn n , having fixed a basis {X 1 , . . . , X n } for g. We have a similar property with the right-invariant vector fieldsX 1 , . . . ,X n . We denote by Diff = ∪ k∈N 0 Diff k the D(G)-module of all the smooth differential operators on G.
The Haar measure is normalised to be a probability measure. It is denoted by dx for integration and the Haar measure of a set E is denoted by |E|.
If f and g are two integrable functions, i.e. in L 1 (G), we define their (non-commutative) The Young's inequalities holds. The convolution may be generalised to two distributions f, g ∈ D (G).
If κ ∈ D (G), we denote by T κ : D(G) → D(G) given via T κ (φ) = φ * κ the associated convolution operator. More generally, in this paper, we will allow ourselves to keep the same notation for a (linear) operator T : D(G) → D (G) and any of its possible extension as a bounded operator on the Sobolev spaces of G since such an extension, when it exists, is unique.

2.2.
Representations. In this paper, a representation of G is any continuous group homomorphism π from G to the set of automorphisms of a finite dimensional complex space. The continuity implies smoothness. We will denote this space H π or identify it with C dπ , where d π = dim H π , after the choice of a basis. We see π(g) as a linear endomorphism of H π or as a d π ×d π -matrix. It is said to be irreducible if the only sub-spaces invariant under G are trivial. If H π is equipped with an inner product (often denoted (·, ·) Hπ , then the representation π is unitary if π(g) is unitary for any g ∈ G. For any representation π, one can always find an inner product on H π such that π is unitary. If π is a representation of the group G, then π(X) = ∂ t=0 π(exp G (X)) defines a representation also denoted π of g and therefore of its universal enveloping Lie algebra (with natural definitions).
If π is a representation of G, then its coefficients are any function of the form x → (π(x)u, v) Hπ . These are smooth functions on G and we denote by L 2 π (G) the complex finite dimensional space of coefficients of π. If a basis {e 1 , . . . , e dπ } of H π is fixed, then the matrix coefficients of π are the coefficients π i,j , 1 ≤ i, j ≤ d π given by π i,j (x) = (π(x)e i , e j ) Hπ . If f ∈ D (G) is a distribution and π is a unitary representation, we can always define its group Fourier transform at π denoted by since the coefficient functions are smooth. If f is integrable, we have One checks easily that the group Fourier transform maps the convolution of two distributions f 1 , f 2 ∈ D (G) to the matrix product or composition of their group Fourier transforms: Two representations π 1 and π 2 of G are equivalent when there exists a map U : H π 1 → H π 2 intertwining the representations, that is, such that π 2 = U π 1 . In this case, one checks easily that L 2 π 1 (G) = L 2 π 2 (G). If π 1 and π 2 are unitary, U is also assumed to be unitary. The dual of the group G, denoted by G, is the set of unitary irreducible representations of G modulo unitary equivalence. We will often identify a representation of G and its class in G.
Theorem 2.1 (Peter-Weyl Theorem). The dual G is discrete. The Hilbert space L 2 (G) decomposes as the Hilbert direct sum ⊕ π∈ G L 2 π (G). Moreover, if for each π ∈ G, one fixes a realisation as a representation with an orthonormal basis of H π , then the functions √ d π π i,j , 1 ≤ i, j ≤ d π , π ∈ G, form an orthonormal basis of G.
The Peter-Weyl theorem yields the Plancherel formula: and the Fourier inversion formula Here C(G) denotes (Banach) space of the continuous functions on G.
We denote by the vector space formed of finite linear sum of vectors in some L 2 π (G), π ∈ G. As each L 2 π (G) is a finite dimensional subspace of D(G), L 2 finite (G) ⊂ D(G). The Peter-Weyl Theorem can be stated equivalently as follows: for any two representations π, π ∈ G, in the sense that d π π i,j (π ) = δ π=π δ i,j for any 1 ≤ i, j ≤ d π , when π is realised as a matrix representation.
, then the Schwartz kernel theorem implies that it is a right convolution operator in the sense that there exists κ ∈ D (G) such that T = T κ : φ → φ * κ on D(G). We have: 2.3. The Laplace-Beltrami operator. Let L be the (positive) Laplace-Beltrami operator of the compact Lie group G, that is, . . , X n are left invariant vector fields which form an orthonormal basis of g. The scalar product comes from the Killing form on the semi-simple part of g and a choice of scalar product on its complement. In this case, L does not depend on a particular choice of such a basis.
The Laplace-Beltrami operator L is a positive self-adjoint operator, having as domain of definition the space of all functions f ∈ L 2 (G) such that Lf ∈ L 2 (G). It is a central operator: it is invariant under left and right translations and its group Fourier transform is scalar: The Peter-Weyl Theorem yields an explicit spectral decomposition for L. Indeed, the eigenvalues of L are {λ π , π ∈ G} and the λ-eigenspace is ⊕ λπ=λ L 2 π (G). Furthermore the 0eigenspace is the space C1 of constant functions on G.
This spectral decomposition shows that for any function f : [0, ∞) → C the operator f (L) is densely defined on L 2 (G). By the Schwartz kernel theorem, it admits a distributional convolution kernel which we denote by f (L)δ 0 ∈ D (G): The group Fourier transform of this kernel is The Sobolev spaces H s (G) = H s may be defined as the Hilbert space which is the closure of D(G) for the norm If s = 0 then H 0 = L 2 (G). If s ∈ N, then H s coincides with the space of function f ∈ L 2 (G) such that Df ∈ L 2 (G) for any D ∈ Diff k , k ≤ s. An equivalent norm is |α|≤s X α · L 2 (G) .
Sketch of the proof of Proposition 2.2. If f ∈ H s , we set f s := (I + L) −s/2 f ∈ L 2 (G) and f s, the orthogonal projection of f s onto ⊕ λπ≤ L 2 π (G) ⊂ L 2 finite (G). Then one checks easily that f := (I + L) −s/2 f s, ∈ ⊕ λπ≤ L 2 π (G) converges in H s to f . The rest of the proof is routine using Lemma A.3.

The symbolic calculus
The operator classes which are the subject of this paper are presented in this section. This requires to also present the notion of difference operators and the symbol classes introduced by Michael Ruzhansky and Ville Turunen [11]. Eventually the main theorem of this paper is stated.
3.1. Symbols and quantisation. The natural quantisation and notion of symbols on Lie groups is due to Michael Taylor [17]. It may be simplified greatly in the case of compact Lie groups, see [11], as G is discrete. In fact, it may be viewed as a generalisation of the Fourier series on tori.
A symbol is a collection σ = {σ(x, π), (x, π) ∈ G × G}. Its associated operator is the operator Op(σ) defined on L 2 finite (G) via For a discussion of the identification of a representation π and its class in G, see [11].
If T is a linear operator defined on L 2 finite (G) (and with image some complex-valued functions of x ∈ G), then one recovers the symbol via (3.1) σ(x, π) = π(x) * (T π)(x), that is, [σ(x, π)] i,j = k π ki (x)(T π kj )(x), when one has fixed a matrix realisation of π. This can be easily checked using (2.5). This shows that the quantisation Op defined above is injective.
Definition 3.1. The symbol σ = {σ(x, π), (x, π) ∈ G× G} admits an associated kernel when for each x ∈ G, we have σ(x, π) ∈ F G (D (G)). Then its associated kernel is κ x := F −1 G σ(x, ·). If κ x is the associated kernel of σ = {σ(x, π), (x, π) ∈ G× G}, the Fourier inversion formula (see (2.4)) implies then for φ ∈ L 2 finite (G), x ∈ G. Remark 3.2. We could have only assume some distributional dependence in x, i.e. the coefficients of x → σ(x, π) are in D (G), then Formula (3.1) would still make sense and be valid. Moreover in this case, by the Schwartz kernel theorem, a sufficient condition for a symbol to admit an associated kernel is that Op(σ)(L 2 finite (G)) ⊂ D (G) and that Op(σ) extends to a linear continuous operator D(G) → D (G), this extension being unique as L 2 finite (G) is dense in D(G) by Proposition 2.2. However in our analysis, we will usually assume regularity in x so we do not seek the greatest generality and prefer assuming that each symbol makes sense at each point x ∈ G. The only exception in this paper is in the proof of Proposition 7.10.
We will usually assume that the symbols are continuous or smooth in the following sense: Definition 3.3. A continuous symbol is a collection σ = {σ(x, π), (x, π) ∈ G × G} such that the associated kernel κ x is a distribution depending continuously on x.
A smooth symbol is a continuous symbol with smooth entries and such that for any D ∈ Diff, {D x σ(x, π)} is a continuous symbol.
If the symbol σ is smooth then x → κ x ∈ D (G) is smooth and Op(σ) : D(G) → D(G) is continuous as an operator valued in D(G).
Example 3.5. For any β ∈ N n 0 , the operator X β admits for symbol π(X) β which does not depend on x. The associated kernel is κ(y) = (X β ) t δ e G (y −1 ).
The following easy lemma shows that one can always approximate an operator of a continuous symbols by an operator with a smooth kernel: Lemma 3.6. Let σ be a symbol. For each ∈ N, we define the symbol σ via Then for a fixed ∈ N, σ admits a kernel κ x ∈ L 2 (G). If σ is continuous or smooth, then so is σ , and for any φ ∈ D(G), we have the uniform convergence on G: Difference operators. The notion of difference operators was introduced by Michael Ruzhansky and Ville Turunen [11]. One motivation comes from the case of the torus, where difference operators are defined naturally. If q ∈ D(G), then we defined the corresponding difference operator ∆ q acting on Fourier coefficients i.e. F G (D (G)) via A carefully chosen family of difference operators will play the role of the derivatives with respect to the Fourier variable in the Euclidean setting. Indeed, if we denote the Euclidean Fourier transform of a (reasonable) function g : R → C by In the Euclidean case, the space of polynomials would yield a 'good' family of difference operators. In the compact case, there is no natural choice. However, we will see that, under certain hypotheses, all the choices are equivalent in a certain sense, see Proposition 4.1.
In order to mimic the choice of difference operators corresponding to polynomials in the Euclidean setting, we need to identify functions with a polynomial behaviour near the neutral element. This is the aim of the following lemma, which follows easily from the structure of Diff k (G) and the Taylor estimates.
(3) There exists a constant C q such that for all x ∈ G, we have |q(x)| ≤ C q |x| a . Definition 3.8. Under the hypothesis of Lemma 3.7, with q satisfying any of the equivalent properties, we say that q vanishes at the neutral element e G up to order a − 1. Moreover if there exists α ∈ N n 0 such that |α| = a and X α q(e G ) = 0, we write [q] := a.
In this case, we say that the order of q or its corresponding difference operator is ∆ q is a.
We extend this to a = 0: a smooth function q is of order 0 if q(e G ) = 0.
Lemma 3.9. Let q ∈ D(G). The order of q is the largest integer a ∈ N 0 such that q(x)/|x| a is bounded on a neighbourhood of e G . Consequently, the mappingq : x → q(x −1 ),q : x → q(x −1 ), and q * : x →q(x −1 ) have the same order as q.
Following the notation of Lemma 3.9, if ∆ is a collection of order-1 difference operators with corresponding functions {q j,∆ } j , we may write∆,∆ and ∆ * for the collections of order-1 difference operators with corresponding functions {q j,∆ } j , {q j,∆ } j and {q * j,∆ } j . We can now define the conditions for an admissible family of difference operators: Definition 3.10. By a collection ∆ of order-1 difference operators, we mean an ordered finite collection of difference operators of order 1. Unless otherwise stated, we will write ∆ = {∆ 1 , . . . , ∆ n ∆ }. Moreover the corresponding functions will then be written as {q 1,∆ , . . . , q n ∆ ,∆ }.
For such a collection ∆ of order-1 difference operators, and we set and that this notation is consistent as any two difference operators commute. We say that ∆ α is of order [q α ∆ ] = |α|. We now define the admissibility of such ∆'s as in [10, Section 2].
Definition 3.11. The collection ∆ of order-1 difference operators is admissible when the gradients at e G of the corresponding smooth functions q 1 , . . . , q n ∆ span the tangent space of G (viewed as a manifold) at e G : rank(∇ e G q 1 , . . . , ∇ e G q n ∆ ) = n (= dim G).
The collection ∆ of order-1 difference operators is said to be strongly admissible when it is admissible and furthermore when e G is the only common zero of the corresponding functions: We can easily construct a strongly admissible collection: The exponential mapping is a diffeomorphism from a neighbourhood of 0 in g onto a neighbourhood of e G . We may assume that this neighbourhood is the ball We fix a basis {x 1 , . . . , X n } of g. For each j = 1, . . . , n, we define a function p j : and then a smooth function q j := p j χ + ψ. The functions q j are of order one and the corresponding collection of difference operators is strongly admissible.
We will need to consider difference operators which satisfy a 'Leibniz-like formula' in the following sense: Definition 3.13. A collection ∆ of order-1 difference operators satisfy the Leibniz-like property when for any Fourier coefficients f 1 and for some coefficients c (j) l,k ∈ C depending only on l, k, j and ∆. Note that this is equivalent to saying that the functions q 1 , . . . , q n ∆ corresponding to ∆ satisfy: Recursively on any multi-index α ∈ N n ∆ 0 , if ∆ satisfies the Leibniz-like property, then for some coefficients c α α 1 ,α 2 ∈ C depending only on α 1 , α 2 , α and ∆, with c α α,0 = c α 0,α = 1. We easily check Lemma 3.14. Let ∆ be a collection of order-1 difference operators. If ∆ is admissible or strongly admissible or with a Leibniz-like property, then so do∆,∆ and ∆ * .
The aim of the rest of this section is to prove: Proposition 3.15. A strongly admissible collection of order-1 difference operators which satisfies the Leibniz-like formula always exists.
In the following lemma, we prove that such a collection of difference operators may be chosen (after a choice of ordering) as the ones associated with the finite family of functions q i,j,π , 1 ≤ i, j ≤ d π , π ∈ G Ad given in the second part of the following lemma. This implies Proposition 3.15.
Lemma 3.16. For each π ∈ G, we fix a unitary realisation acting on C dπ and denote by π i,j , 1 ≤ i, j ≤ d π the corresponding matrix entries. We also set Then all the functions q i,j,π , 1 ≤ i, j ≤ d π , π ∈ G, are smooth, vanish at e G and satisfy: The rank of the matrix (∇ e G q i,j,π ) 1≤i,j≤dπ is not zero for any π ∈ G\{1}. Moreover there exists a finite subset denoted by G Ad of G\{1} such that the only common zero of the functions q i,j,π , 1 ≤ i, j ≤ d π , π ∈ G Ad is e G , and that the rank of the matrix (∇ e G q i,j,π ) 1≤i,j≤dπ,π∈ G Ad is n = dim G.
The proof of Lemma 3.16 essentially follow the arguments given in Proposition 4.3 and Lemma 4.4 of [10].
Proof of Lemma 3.16. The functions q i,j,π defined in the statement are smooth and vanish at 0 as π i,j (e G ) = δ i,j . The formula for q i,j,π (xy) follows readily from the equality π(xy) = π(x)π(y) expressed in coordinates. If for one π ∈ G, all the tangent vectors ∇ e G q i,j,π are zero, this means that π viewed as the infinitesimal representation of the Lie algebra g is identically zero and thus that π is trivial. Hence the first part of the statement is proved.
To prove the last part of the statement, let us first assume that the centre of the group G is trivial. Let G Ad be the (class of) non-trivial representations intervening in the decomposition (possibly with multiplicities) of the adjoint representation of G on its Lie algebra g. The kernels of the adjoint representation of G and the (infinitesimal) adjoint representation of the Lie algebra g being trivial implies easily the two properties. Now let us assume that the group G admits a non-trivial centre Z(G). Then we may view G as the direct product of Z(G) with G 1 = G/Z(G) and an element of G as the tensor product of an element of Z(G) with an element of G 1 . One can then choose as G Ad the collection of the representations 1 Z(G) ⊗ π 1 , π 1 ∈ ( G 1 ) Ad , together with χ ⊗ 1 G 1 where χ runs over a set of (non-trivial) generators of the morphisms in Z(G).

3.3.
Symbol and operator classes. The symbol classes considered in this paper were introduced by Michael Ruzhansky an Ville Turunen in [11]. Convention: In this paper, ρ and δ are two real numbers satisfying and β ∈ N n 0 there exists C > 0 such that As the group G is compact and σ is smooth in x, σ ∈ S m ρ,δ (G, ∆) if and only if for any D ∈ Diff b and any α ∈ N n ∆ 0 . However, it may be convenient to check the condition only for {X β , β ∈ N n 0 }. For instance, this allows us to define for a, b ∈ N 0 : If x ∈ G is fixed (and if there is no ambiguity), we may use the notation Note that if a symbol has smooth entries and satisfies (3.4) then σ is a smooth symbol in the sense of Definition 3.4, and the operator Op(σ) is a continuous operator D(G) → D(G), see Section 3.1.
We denote by Ψ m ρ,δ (G, ∆) the corresponding operator classes: One checks easily that if then the inclusion S m 1 ρ 1 ,δ 1 ⊂ S m 2 ρ 2 ,δ 2 holds and is continuous. Hence the same is true for the operator spaces Ψ m 1 ρ 1 ,δ 1 ⊂ Ψ m 2 ρ 2 ,δ 2 . We say that a symbol or an operator is smoothing when it is in S −∞ (G, ∆) = ∩ m∈R S m ρ,δ (G, ∆). One checks easily that it does not depend on ρ and δ.
Note that the calculus is invariant under translations in the following sense: xo has κ xox as kernel and σ(x o x, π) as symbol, and The main result. We can now state the main result of this paper.
(1) For each m ∈ R, the symbol class S m ρ,δ (G, ∆) defined in Definition 3.17 and its corresponding operator class Ψ m ρ,δ (G, ∆) are independent of the choice of a strongly admissible collection ∆. They are denoted by S m ρ,δ and Ψ m ρ,δ respectively. Some aspects of Theorem 3.19 were announced in [11] and [10], essentially the properties of composition and adjoint for when ρ = δ, the boundedness on L 2 (G) for symbol of order 0 and any ρ, δ. However one may argue that many of the proofs in [11] and [10] were incomplete or barely sketched. For instance the composition property (which intervenes in most of the subsequent proofs) is not proved, see the introduction. The technical purpose of this article is to show completely the fundamental properties stated in Theorem 3. 19.
For simplicity, we will allow ourselves to refer to ∪ m∈R Ψ m ρ,δ as 'the' calculus, although it is the object of this paper to show that it is a pseudo-differential calculus.
Another important result of the paper is the fact that the Laplace operator and its spectral calculus are part of the calculus. Proposition 3.20. Let ∆ be a strongly admissible collection of order-1 difference operators. For any m ∈ R and muti-index α ∈ N n ∆ 0 , there exists C > 0 such that for all for any function We will also need a property as in Proposition 3.20 but for multipliers in tL, uniformly in t ∈ (0, 1): Proposition 3.21. Let ∆ be a strongly admissible collection of order-1 difference operators. For any m ∈ R and muti-index α ∈ N n ∆ 0 , there exists d ∈ N 0 and C > 0 such that for all in the sense that if the supremum in the right hand-side is finite, then the left hand-side is finite in the inequality holds.
The proofs of Propositions 3.20 and 3.21 are provided in A. Proposition 3.21 is the main technical argument of this paper. It enable us to use Littlewood-Payley decompositions and analyse precisely the singularity of the kernels, and these two results are the keys to show the rest of the properties of the calculus.
More precisely, the proof of Theorem 3.19 proceeds as follows. In Section 4, we show that the symbol classes form an algebra independent of the choice of a strongly admissible collection (this proves Part (1) of Theorem 3.19). We also show that the differential calculus is in the calculus, this shows Part (3) of Definition 1.1. Note that the inclusions in Part (1) of Definition 1.1 follow easily from the definition of the symbol classes. In Section 5, we show estimates for the kernels of the operators in the calculus. In Section 6, we prove the properties for the adjoint and the composition, that is, Parts (2) and (3) of Definition 1.1. We will also obtain the usual properties of asymptotic expansions in the case ρ = δ. In Section 7, we show that pseudo-differential operators are bounded on Sobolev spaces and this concludes the proof of the calculus' fundamental properties in the sense of Definition 1.1. Consequently Part (2) of Theorem 3.19 is proved. In Section 7, we also give a commutator characterisation of the operators in the calculus. Beals' commutator characterisation [3] localised to manifolds then implies that the calculi studied here coincides with Hörmander's and Part (3) of Theorem 3.19 holds.

First results
In this section, we show that the symbol classes form an algebra independent of the choice of a strongly admissible collection of order-1 difference operator and that the differential calculus is in the calculus.
4.1. ∆-equivalence. We can already prove Theorem 3.19 (1), that is, that the symbol classes do not depend on a choice of ∆'s in the following sense: Proposition 4.1. If ∆ and ∆ are two strongly admissible collections of order-1 difference operators, then for any m ∈ R and 1 ≥ ρ ≥ δ ≥ 0, the Fréchet spaces S m ρ,δ (G, ∆) and S m ρ,δ (G, ∆ ) coincide, that is, the vector spaces together with their topologies coincide.
The proof of Proposition 4.1 relies on the following lemma: Let q, q ∈ D(G) be two functions which do not vanish except maybe at e G and such that q/q extends to a smooth function on G. Let s ∈ R and let σ = {σ(π)} be a symbol such that Then we have the same property for ∆ q σ with the same s. More precisely, there exists C = C q,q ,s > 0 (independent of σ) such that Proof of Lemma 4.2. Let q, q as in the statement. Let κ ∈ D (G) and s ∈ R. We have to prove Let φ ∈ D(G). We have having used the Sobolev inequalities (cf. Lemma A.3). We have obtained One can see easily that and thus by duality and interpolation, we also have the same property for any s ∈ R, with the slight modification that the maximum is now over |α| ≤ |s| + 1. Hence in our case, we obtain that We have obtained (4.1).
Proof of Proposition 4.1. Let ∆ be a strongly admissible collections of order-1 difference operators with corresponding functions q 1 , . . . , q n ∆ . Up to reordering ∆, we may assume that the rank of (∇ e G q 1 , . . . , ∇ e G q n ) is n = dim G. Furthermore the basis of g is chosen to be (X 1 , . . . , X n ) = (∇ e G q 1 , . . . , ∇ e G q n ). For each q j , j = 1, . . . , n, we use the notation of Lemma 3.12 to construct q j,0 := p j χ + ψ. We adapt the argument of Lemma 3.12 for the other functions. That is for j > n, we know that ∇ e G q j may be written as a linear combination n =1 c (j) ∇ e G q and we define then if y ∈B( 0 ), and q j,0 := p j χ + ψ.
Let ∆ 1 and ∆ 2 be two collections constructed in Lemma 3.12 out of two bases (X (1) j ) and (X (2) j ) of g. Let P be a n × n real matrix mapping (X j ). We construct the two corresponding collections of functions (q (1) j ) and (q (2) j ) as in Lemma 3.12. We check easily that for each j, ( k P j,k q are smooth on G. By Lemma 4.2, the Fréchet spaces S m ρ,δ (G, ∆ 1 ) and S m ρ,δ (G, ∆ 2 ) coincide for each m, ρ, δ. Hence S m ρ,δ (G, ∆) do not depend on a choice of strongly admissible collection ∆.

Remark 4.3 (Convention)
. When writing S m ρ,δ (G, ∆) or Ψ m ρ,δ (G, ∆), we assume that a collection of order-1 difference operator has been fixed. From now on, as in Theorem 3.19 (1), when we write S m ρ,δ and Ψ m ρ,δ , we mean the symbol class S m ρ,δ (G, ∆) and its corresponding operator class Ψ m ρ,δ (G, ∆) for any choice of a strongly admissible collection ∆. This is justified by Proposition 4.1. We can always assume that this collection also satisfies the Leibniz-like property, see Proposition 3.15.
4.2. The differential calculus. In this subsection, we prove that the differential calculus, that is, ∪ m∈N 0 Diff m , is included in the calculus.

4.3.
The algebra of symbols. In this section, we summarise almost immediate properties of the classes of symbols. Let 1 ≥ ρ ≥ δ ≥ 0.

Consequently, we have:
Corollary 4.6. The classes of symbols ∪ m∈R S m ρ,δ form an algebra. Furthermore if σ 0 is smoothing, then for any σ ∈ S m ρ,δ , the symbols σσ 0 and σ 0 σ are also smoothing.

The associated kernels
In this section, we show that the kernels of the symbols we have considered can only have a singularity at the neutral element and we obtain estimates near this singularity. We also show that these distribution may be approximated by smoother kernels.

5.1.
Approximations by nice kernels. We have already seen that the kernel associated with a continuous symbol can be approximated by a smooth kernel in the sense of Lemma 3.6. It will be useful to have a more precise version for the symbols in S m ρ,δ .

5.2.
Singularities of the kernels. Let us show that the singularities of the convolution kernels in Ψ ∞ ρ,δ can be located only at the neutral element in the following sense: Proposition 5.2. We consider the symbol class S m ρ,δ (G, ∆) with 1 ≥ ρ ≥ δ ≥ 0, ρ = 0, and a collection ∆ such that if ∩ q∈∆ {x ∈ G : q(x) = 0} = 0.
If σ ∈ S m ρ,δ , then its associated kernel (x, y) → κ x (y) is smooth on G × (G\{e G }). If σ ∈ S −∞ is smoothing, then its associated kernel (x, y) → κ x (y) is smooth on G × G. The converse is true.
The proof relies on the following lemma and its corollary: in the sense that κ ∈ L 2 (G) when there exists s > n/2 such that the right-hand side is finite.
The properties of the Hilbert-Schmidt operators and the Plancherel formula yield Corollary 5.4. If σ ∈ S m ρ,δ with 1 ≥ ρ ≥ δ ≥ 0 and ∆ a collection of order-1 difference operators, then for any differential operators D z ∈ Diff b and D x ∈ Diff b , the function D x D z {q α ∆ (z)κ x (z)} is continuous on G and bounded, up to a constant of m, ρ, δ, ∆, b, b by sup π∈ G σ(x, π) S m ρ,δ (G,∆),|α|,b as long as b + m + n + δb < ρ|α|. Proof. If s ∈ R, using the properties of the Sobolev spaces, we have:

5.3.
Estimates for the kernel. In this section , we study the behaviour of the kernels near the origin. More precisely, we show: Proposition 5.5. Let σ ∈ S m ρ,δ with 1 ≥ ρ ≥ δ ≥ 0, ρ = 0, and ∆ strongly admissible. Then its associated kernel (x, y) → κ x (y) ∈ C ∞ (G × (G\{e G }) satisfies the following estimates: • if n + m > 0 then there exists C and a, b ∈ N (independent of σ) such that • if n + m = 0 then there exists C and a, b ∈ N (independent of σ) such that • if n + m < 0 then κ x is continuous on G and bounded Using the properties explained in Section 4.3, we also obtain similar properties for any derivatives in x and y of κ x (y) multiplied by a smooth function q.
First we need to understand a 'dyadic piece' of a symbol in the calculus: For any t ∈ (0, 1) we define the symbol σ t via σ t (x, π) := σ(x, π)η(tλ π ). Then for any m 1 ∈ R we have where C = C m,m 1 ,a,b,η does not depend on σ or t ∈ (0, 1).
Proof of Lemma 5.6. We may assume that the collection ∆ of order-1 difference operators satisfies the Leibniz-like property described in Definition 3.13 (see Remark 4.3). Using the Leibniz-like property and then Proposition 3.21 for any m 2 ∈ R yields Choosing m 2 = m 1 − m, the statement follows readily.
We suppose that a strongly admissible collection ∆ has been fixed. Applying Corollary 5.4 and its proof for any α ∈ N n 0 (but no derivatives), for any m 1 ∈ R, whenever m 1 + n < ρ|α| we have by Lemma 5.6. The strong admissibility implies ∀z ∈ G, a ∈ N 0 , |z| a ∆,a |α|=a |q α (z)|.
Hence for any a ∈ N 0 and m ∈ R satisfying m 1 + n < ρa, we have obtained: We may assume |z| < 1 and choose 0 ∈ N 0 such that Case of m + n > 0. For ≤ 0 , we choose the real number m 1 ∈ R and the integer a ∈ N 0 to be such that so |κ x (z)| σ S m ρ,δ ,a,0 (1 + | ln |z||) σ S m ρ,δ ,a,0 | ln |z||. This shows the statement in the case m + n = 0 and concludes the proof of Proposition 5.5.

The calculus
In this section, we prove that ∪ m∈R Ψ m ρ,δ satisfies the algebraic properties of a calculus, that is, Parts (1), (2) and (3) of Definition 1.1. Clearly Part (1) holds, see Section 3.3. It remains to prove Parts (2) and (3), that is, the properties for the adjoint and the composition. We will also obtain the usual properties of asymptotic expansions in the case ρ = δ.
6.1. Asymptotic expansions. The analysis to prove the properties for the adjoint and the composition will also yield a familiar (but matrix valued) expansion. This section is devoted to understand the meaning of the expansion and the coefficients in it.
For any ∈ N 0 , the sum is finite. As S m 0 ρ,δ is a Fréchet space, we obtain that σ = ∞ j=0σ j is a symbol in S m 0 ρ,δ . Starting the summation at j = M + 1, the same proof gives ∞ j=M +1σ j ∈ S m M +1 ρ,δ . Hence the symbol given via as the symbol (1 − ψ)(t j λ π ) is smoothing by Proposition 3.21 and so is σ j (1 − ψ)(t j λ π ), see Section 3.3.
The property in (6.1) is proved but it remains to show that the symbol σ is unique modulo smoothing operator. If τ is another symbol as in the statement of the theorem, then for any In the expansion given for adjoint and composition, we will need to identify a suitable choice of ∆ together with a choice of vector fields. This is the purpose of the next lemma, whose proof is left to the reader: Lemma 6.2. Let ∆ be a strongly admissible collection of order-1 difference operators and q 1,∆ , . . . , q n ∆ ,∆ the corresponding smooth functions. We may assume that n ∆ = n.
There exists an adapted basis X ∆ := X ∆,1 , . . . , X ∆,n such that X j {q(· −1 )}(e G ) = δ j,k . The following Taylor estimates hold for any integer N ∈ N 0 and y ∈ G: where the constant C > 0 depends in N, G, ∆ but not on f ∈ D(G). Furthermore for any β ∈ N n 0 , we have on the one hand x,N and on the other hand, Here and in the rest of the paper, if N ∈ N 0 , then R f x,N denotes the Taylor remainder of f at x of order N − 1 (adapted to the fixed collection ∆): and X α ∆ = X α 1 ∆,1 . . . X αn ∆,n . If N < 0 then R f x,N ≡ f (x ·).
Proof. The proof is straightforward. The properties of the remainder follow from the facts that left and right invariant vector fields commute and that the Taylor expansion is essentially unique.

Adjoint. This section is devoted to showing
Proposition 6.3. If T ∈ Ψ m ρ,δ then its formal adjoint T * is also in Ψ m ρ,δ .
In fact we will also prove that T → T * is continuous on Ψ m ρ,δ . Hence, by Lemma 5.1, we may assume that all the associated kernels are smooth on G × G. This justifies the following formal manipulations. One computes easily that if T = Op(σ) ∈ Ψ m ρ,δ with associated kernel κ x then T * = Op(σ ( * ) has associated kernel κ ( * ) x given by κ ( * ) x (y) =κ xy −1 (y −1 ).
Note that this κ ( * ) x is usually different from the kernel κ * x : y →κ x (y −1 ) associated with σ * (unless, for instance, the symbol does not depend on x) but we have κ ( * ) x (y) = κ * xy −1 (y). In this section, we assume that ∆ and X ∆ are fixed and chosen as in Lemma 6.2. We also simplify slightly the notation by setting X ∆,j = X j . Lemma 6.4. Let σ ∈ S m ρ,δ and let κ x be its associated kernel. We assume that (x, y) → κ x (y) is smooth on G × G. Then for any multi-indices β, β 0 , α 0 ∈ N n 0 , there exists N 0 ∈ N 0 such that for any integer N > N 0 , we have where the constant C > 0 and the semi-norm · S m ρ,δ ,a,b are independent on σ (but may depend on N, m, ρ, δ, ∆, α 0 , β 0 , β).
Proof of Lemma 6.4. The idea is to use the estimate given in Lemma 6.2 for the Taylor reminder in the case β = β 0 = α 0 = 0. More generally, for any multi-indices, using (2.1), we have: assuming N > |β|. We apply Proposition 5.5 (see also Section 4.3) to estimate the maximum: with e = n + m + δ(|β 0 | + |α|) + |β 1 | − ρ|α 0 |. Thus the sum in (6.3) is This is integrable when the following implication holds We choose N 0 ∈ N such that N 0 > |β| + 1 is the smallest integer satisfying the implication just above. This shows Lemma 6.4.
We will need the following very crude lemma, which follows readily from the properties of the left or right invariant vector fields, especially (4.2). The proof is left to the reader. Lemma 6.5. Let · S m ρ,δ ,a,b be a semi-norm. Then there exists C > 0 such that for any smooth symbol σ, we have (with possibly unbounded quantities) Proof of Proposition 6.3. Let σ ∈ S m ρ,δ . First we assume that its associated kernel (x, y) → κ x (y) is smooth on G × G. We set and apply Lemma 6.5, We see that τ N (x, ·) is the group Fourier transform of y → R κ * · (y) x,N (y −1 ) given in (6.2). Using (2.6) and (2.2), we see that each maximum above is bounded by the integral given in Lemma 6.4. Thus for N ≥ N 0 with N 0 , a , b depending on m, ρ, δ, a, b, we have . From the properties of the symbol classes (see Section 4.3), the sum |α<N ∆ α X α x σ(x, π) * is a symbol in S m ρ,δ . This implies that σ ( * ) is also in S m ρ,δ and depend continuously on σ. By Lemma 5.1, this extends to any symbol σ.

Composition. This section is devoted to showing
Proposition 6.7. If T 1 ∈ Ψ m 1 ρ,δ and T 2 ∈ Ψ m 2 ρ,δ , then the composition We proceed in a similar way as in Section 6.2. In this section, we assume that ∆ and X ∆ are fixed and chosen as in Lemma 6.2. We also simplify slightly the notation by setting X ∆,j = X j .
One computes easily that if T i = Op(σ i ) ∈ Ψ m ρ,δ , with associated kernel κ i,x i = 1, 2, (which we assume smooth on G × G) then T 1 T 2 has associated kernel κ x given by Note that this κ x is usually different from κ 2x * κ 1x , unless, for instance, σ 2 does not depend on x.
Proof of Lemma 6.4. We notice that thus taking the group Fourier transform x,N (z −1 )dz having used the notation for the Taylor estimate for a matrix valued function -which is possible. We may assume, and we do, that the collection of difference operators corresponding to the q j satisfies the Leibniz like property, see Remark 4.3. Using this and the Leibniz property for vector fields, one checks easily that z π(z) * and using integration by parts. Using the Taylor estimates, see Lemma 6.2, we have By Proposition 5.5 (see also Section 4.3), we have if e < 0 with e = n + m + δ|β 0,1 | + |β 1 | − ρ|α 1 |. Thus the last sum in (6.5) is where I is the integral This is integrable, that is I < ∞ when N > 1 + 2b and the following implication holds We choose N 0 ∈ N such that N 0 > 1 + 2b is the smallest integer satisfying the implication just above. This shows Lemma 6.4.

Boundedness on Sobolev spaces and commutators
In this section, we show that pseudo-differential operators are bounded on Sobolev spaces and we give a commutator characterisation of the operators in the calculus. 7.1. Boundedness on L 2 (G). This section is devoted to showing that operators of order 0 are bounded on L 2 (G) in the following sense: Moreover the constant C may be chosen of the form C = C σ S 0 ρ,δ ,a,b with C > 0 and · S 0 ρ,δ ,a,b independent of σ (but maybe depending on G and ρ, δ). Given the continuous inclusions of the spaces S 0 ρ,δ , it suffices to prove the case ρ = δ. We first show the case ρ = δ = 0 and then the case ρ = δ (strictly) positive.
The case (ρ, δ) = (0, 0) follows from the following easy lemma: Lemma 7.2. If σ is a smooth symbol such that then Op(σ) is bounded on L 2 (G): Moreover the constant C may be chosen of the form C = C C 0 with σ independent of σ. Moreover if δ > 0, then The proof of Lemma 7.2 follows easily from the Sobolev embedding (cf. Lemma A.3): Proof of Lemma 7.2. Let T = Op(σ), σ ∈ S 0 0,0 and f ∈ D(G). Sobolev's inequalities yield As X α z f * κ z (x) = T Xzκz (f ), after integration over G, we obtain: We conclude with C 0 = max T X α z κz L (L 2 (G)) , z ∈ G, α| ≤ n 2 .
We now show the case ρ = 0.
We see that T j T * k = T (η j η k )(L)T * is identically zero if |j − k| > 2. If |j − k| ≤ 2, we use Lemma 7.2 (see also Remark 7.3), and then the properties of composition and adjointness (cf. Propositions 6.7 and 6.3): 4 . As (I + L) m /2 ∈ Ψ m , the second semi-norm above is finite. By Lemma 5.6, the semi-norms of T j and T k are bounded uniformly in the same semi-norm in σ. Hence we have obtained For T * j T k , as in the previous paragraph, we use Lemma 7.2 and Propositions 6.7 and 6.3). But we also introduce powers of (I + L) By Lemma 5.6, the first and third semi-norms above are 2 −j c 2 and 2 k c 2 . The second seminorm above is σ S 0 ρ,ρ ,a 4 ,b 4 . Choosing c to be the sign of k − j, this shows that {T j } j∈N 0 satisfies the hypotheses of the Cotlar-Stein Lemma [15, Section VII.2] and concludes the proof of Lemma 7.4.
This also concludes the proof of Proposition 7.1. We obtain the continuity on (L 2 -)Sobolev spaces with loss of derivatives controlled by the order: Corollary 7.5. Let 1 ≥ ρ ≥ δ ≥ 0 with δ = 1 and m ∈ R. If σ ∈ S m ρ,δ , then Op(σ) maps boundedly the Sobolev spaces H s → H s−m for any s ∈ R and we have where the constant C > 0 and the semi-norm · S m ρ,δ ,a,b are independent of σ (but may depend on s, m, ρ, δ, G).
Another consequence is the continuity for commutators, see the next section. We will need the following property: Lemma 7.6. Let 1 ≥ ρ ≥ δ ≥ 0 with δ = 1, ρ = 0, and m ∈ R. If q is a smooth function on G vanishing up to order a 0 ∈ N 0 . and if σ ∈ S m ρ,δ , then Op(∆ q σ) maps H m−ρa 0 boundedly to L 2 (G) and where the constant C > 0 and the semi-norm σ S m ρ,δ ,a,b are independent of σ (but may depend on q, a 0 , m, ρ, δ, ∆, G).
Proof of Lemma 7.6. Let χ ∈ D(G) be valued in [0, 1] and such that χ| B( 0 /2) ≡ 1 and χ| B( 0 ) c ≡ 0. We write ∆ q σ = ∆ qχ σ + ∆ q(1−χ) σ. As the kernel associated with ∆ q(1−χ) σ is smooth, this symbol is smoothing. Let ∆ be a strongly admissible collection of order-1 difference operators and {q j } n j=1 the corresponding functions, for instance the ones constructed in Lemma 3.12. It is not difficult to construct a smooth function q as a linear combination of q α = q α 1 1 . . . q αn n , |α| = a, such that χq/q is smooth on G. We check easily that where ψ x (y) = χq/q (y −1 x), thus by the Sobolev embedding (cf. Lemma A.3), We argue in a similar way as at the end of the proof of Lemma 4.2 to obtain and we conclude with by Corollary 7.5.

7.2.
Commutators. We adopt the following notation: if q ∈ D(G) and D ∈ Diff, we denote by L q and M D the commutators defined via • If q is a smooth function and if T : More generally, for any s 1 , s 2 ∈ R, we have L q T L (H s 1 ,H s 2 ) ≤ 2C q,s 1 ,s 2 T L (H s 1 ,H s 2 ) since max( qT L (H s 1 ,H s 2 ) , T q L (H s 1 ,H s 2 ) ) ≤ C q,s 1 ,s 2 T L (H s 1 ,H s 2 ) .
Proof of Lemma 7.7. The first part is easily checked by direct computations. The second part follows from the continuity of φ → qφ on any H s for any q ∈ D(G).
The Leibniz properties yield: (1) Let ∆ be a collection of order-1 difference operators satisfying the Leibniz-like property as in Definition 3.13, and q 1 , . . . , q n ∆ the corresponding smooth functions. Then, for any continuous symbol σ, we have: with the same coefficients c (j) l,k ∈ C as in Definition 3.13, andq j (x) = q j (x −1 ).
(2) For any X ∈ g and any smooth symbol σ, we have Proof. For the first formula, we apply (3.3) to q j (x) = q j (y y −1 x) in For the second formula, we apply (3.3) to q j (y −1 x) in We write Op(σ)q l = (q l − Lq l )Op(σ) and observe that 1≤l,k≤n ∆ c (j) l,k q kql = −(q j +q j ), having applied (3.3) to x, y = x −1 . Thus we obtain: For the second part, we seẽ If ∆ is a collection of order-1 difference operators or equivalently a collection of smooth functions {q 1 , . . . , q n ∆ } and if X 1 , . . . , X n form a basis of g, then we set Proposition 7.9. Let 1 ≥ ρ ≥ δ ≥ 0 with δ = 1 and m ∈ R. If T ∈ Ψ m ρ,δ , then L α M β T extends boundedly in an operator from H m−ρ|α|+δ|β| to L 2 (G) for each α ∈ N n ∆ 0 , β ∈ N n 0 and for L α ∆ , M β ∆ as defined in (7.1) where ∆ is any collection of order-1 difference operators.
where the constant C > 0 and the semi-norm · Ψ m ρ,δ ,a,b are independent on T (but may depend on α, β, ∆ and the choice of basis for g).
If ∆ satisfies a Leibniz-like property, then Corollary 7.5 and Lemma 7.8 imply Proposition 7.9. In the general case, we have to use Lemma 7.6 and the ideas of its proof.
Recall that a linear operator T : D(G) → D (G) is in Ψ m ρ,δ (G, loc) when for any φ, ψ ∈ D(G) supported in charts of G, the operator φT ψ : f → φT (ψf ) viewed as an operatorT φ,ψ on R n is in Ψ m ρ,δ (R n ). The hypotheses on ρ and δ, that is, 1 ≥ ρ > δ ≥ 0, ρ ≥ 1 − δ, ensure that the operators in Ψ m ρ,δ (G, loc) are well defined using changes of charts. Proof of Corollary 7.12. Let T ∈ Ψ m ρ,δ . Let also φ, ψ ∈ D(G) supported in charts of G. By Lemma 3.18 and the linearity of T , we may assume that ψ is supported in the 'small' neighbourhood B( 0 /2) of e G and use the exponential mapping there as chart. We apply Corollary 7.11 with a basis of right invariant vector fields and the collection ∆ constructed in Lemma 3.12. This implies thatT φ,ψ satisfies the hypotheses of Beal's characterisation of pseudo-differential operators (for the commutators of ∂ x i and x j ) [3]. ThusT φ,ψ ∈ Ψ m ρ,δ (R n ) and T ∈ Ψ m ρ,δ (G, loc). The converse holds for the same reasons.

Appendix A. Multipliers of the Laplace operator
This appendix is devoted to the proofs of Propositions 3.21 and 3.20. They rely on (now classical) methods to estimate weighted norms of kernels of spectral L-multipliers using the heat kernels.
We reformulate Proposition 3.21 into the following property: Proposition A.1. For any m ∈ R, q ∈ D(G) vanishing nowhere except maybe at e G , there where the constant C may be chosen as C f M m/2 ,d with C depending only on m, q and the group G but not on f, t, π.
In the statement above, we have used the following notation for d ∈ N 0 and m ∈ R: Proposition A.1 easily implies: For any m ∈ R and q ∈ D(G) vanishing nowhere except maybe at e G , there exists C such that for any function f : SpecL → C satisfying sup λ∈SpecL (1 + λ) − m 2 |f (λ)| < ∞ and π ∈ G, we have Proof. We can construct the functioñ where the functions φ µ ∈ D(R) are bump functions valued in [0, 1] with disjoint supports and such that φ µ (µ) = 1. We havef ∈ C ∞ [0, ∞), f (L) =f (L) and Hence we may assume f =f ∈ C ∞ [0, ∞).
More precisely, we can choose the bump functions as where χ ∈ D(R) is a fixed function such that 2 ] ≡ 1, and where δ 0 := min{|λ 1 − λ 2 |, λ 1 = λ 2 ∈ SpecL} is the minimum distance between two distinct eigenvalues of L. In this case, we have We then apply Proposition A.1 tof and, for instance, t = 1.
Corollary A.2 easily implies the first and second part of Proposition 3.20. The last part follows from the following remark: it is possible to extend the proof presented in this appendix to symbols depending on x in the following way: σ(x, π) = f (x, λ π ), for a function f very regular in x ∈ G.
Hence this section is devoted to the proof of Proposition A.1, which will be presented in A.4. Before this, we present its main tool, the heat kernel, whose properties will be recalled in A.1. We also state and prove technical lemmata in A.2 and A.3.
A.1. The heat kernel. The heat kernel, i.e. the kernel of the operator e −tL : is a positive smooth function on G which satisfies ∀s, t > 0 G p t (x)dx = 1, p t (x −1 ) = p t (x), and p t * p s = p t+s . and the following estimates [19] In these estimates, C is independent of x ∈ G and t > 0 but may depend on the multi-index α ∈ N n 0 . V (r) denotes the volume of the ball centred at e G and of radius r > 0. It may be estimated via and [19, p.111] For the sake of completeness, let us sketch the proof of the following well known facts: Lemma A.3. If s > n/2, then the kernel B s of the operator (I + L) −s/2 is square integrable and the continuous inclusion H s ⊂ C(G) holds.
Sketch of the proof of Lemma A.3. If s > 0, the properties of the Gamma function and of the heat kernel together with the spectral calculus of L imply that the kernel B s of the operator (I + L) −s/2 is the integrable function given via: and that we have It is not difficult to show that this last integral against dt 1 dt 2 is finite whenever s > n/2. The Sobolev embedding then follows easily from the fact that one can write f = {(I+L) −s/2 f } * B s for any f ∈ H s with s > n/2.
A.2. Technical lemmata. This section is devoted to stating the following property for compactly supported functions which is the main step in the proof of Proposition A.1: (1) Let q ∈ D(G) and m ∈ R. There exists C = C q,m such that for any continuous function f with support in [0, 2], we have for any t ≥ 0 (2) For any q ∈ D(G), m ∈ R and β ∈ N n 0 , there exists C = C q,m,β and d = d a,m,β ∈ N such that for any function f ∈ C d [0, ∞) with support in [0, 2], π ∈ G we have for any t ∈ (0, 1) Remark A.5.
(1) It is not difficult to prove that, if f is compactly supported in [0, ∞), then the kernel of f (L) is smooth and thus the integrals intervening in Lemma A.4 are finite. Indeed this follows readily from SpecL ⊂ [0, ∞) being discrete and the fact that the eigenspaces of L are finite dimensional and included in D(G). However Lemma A.4 yields bounds for these integrals in terms of f and t which will be useful later.
(2) The second part of Lemma A.4 implies that for any q ∈ D(G), β, γ ∈ N n 0 , we have: with the constant C = C q,β,γ > 0 independent of f . This follows easily from and [X γ 1 q] ≤ [q] − |γ| and φ is any reasonable function on G.
The two following lemmata will be useful in the proof of Lemma A.4 given in the next section.
Lemma A.6. For any q ∈ D(G), there exists C = C q such that for any r > 0 we have Proof of Lemma A.6. We can estimate directly q L 2 (B(r)) ≤ q ∞ . If r is small, we can obtain a better estimate using Lemma 3.7 (3) and the fact the ball B( 0 ) yields a chart around the neutral element. More precisely we have The second lemma is a classical construction.
Lemma A.7. Let g ∈ S(R) be an even function such that its (Euclidean) Fourier transform satisfies: Such a function exists. For any d ∈ N and any h ∈ S (R) satisfying h ∈ C d (R) with h (d) ∞ < ∞, we have where g δ is the function given by g δ (x) = δ −1 g(δ −1 x).
Sketch of the proof of Lemma A.7. The hypothesis on g implies R g(x)dx = 1 and R x g(x)dx = 0 for all ∈ N.
Using the Taylor formula on h, we have where R d (x, ·) is the Taylor remainder of the function h at x of order d. We conclude easily with the following (x-independent) estimate for the remainder: A.3. Proof of Lemma A.4. This section is devoted to proving Lemma A.4. We will use the classical technics relying on estimates for the heat kernel, see [19,1]. More precisely, we will follow closely the presentation of [8].
We fix a function f : [0, ∞) → C with compact support in [0,2]. We assume that f is regular enough, more precisely in C d [0, ∞), that is, d-differentiable with d-th continuous derivatives. d will be suitably chosen.
Step 1: For each t > 0, we define the function h t : [0, ∞) → C via The spectral theorem implies easily For the L 2 -norm of the heat kernel, we use (A.1) and (A.4) to obtain This implies f (tL)δ 0 ∈ L 2 (G) with the following estimate: Step 2: Let us show that the integral in the statement on a ball of radius √ t near the origin may be estimated by: |q(x) f (tL)δ 0 (x)|dx q min(1, t [q] 2 ) f ∞ .
In both cases i = 1, 2, we will use Cauchy-Schwartz' inequality t,j L 2 (A t,j ) .
For the first L 2 -norm, we use Lemma A.6 (with t small): Step 4a: For the second L 2 -norm, in the case i = 2, we have M (2) t,j L 2 (A t,j ) ≤ M (2) t,j L 2 (G) ≤ h t ( √ L) L (L 2 (G)) X β p t 1 B(2 j−1 √ t) c L 2 (G) . On the one hand, we have by the spectral theorem On the other hand, the estimate for the heat kernel in (A.2) yields Collecting the previous estimates yields: The exponential decay allows us to sum up over j and to obtain: Step 4b: The case of i = 1, that is, the estimate of M (1) t,j L 2 (A t,j ) , requires a more sophisticated argument. The function h t is even and has compact support. Assuming f ∈ C d [0, +∞) with d ≥ 2, the function h t ∈ C d (R d ) admits an integrable Euclidean Fourier transform of h t ∈ L 1 (R). Hence the following formula holds for any µ ∈ R h t (µ) = 1 2π R cos(sµ) h t (s)ds, µ ∈ R, The operator cos(s √ L) has finite unit propagation speed [18,ch. IV] in the sense that supp{cos(s √ L)δ 0 } ⊂ B(|s|). This implies x ∈ A t,j and |s| ≤ 2 j−1 √ t =⇒ cos(s √ L) X β p t 1 B(2 j−1 √ t) (x) = 0.
We use this property in the following way. Let g ∈ S(R) and g δ = δ −1 g(δ −1 ·) be functions as in Lemma A.7. As supp g (2 j−1 √ t) −1 ⊂ [−2 j−1 √ t, 2 j−1 √ t], the finite propagation speed property implies Hence we can rewrite (A.11) for any x ∈ A t,j as , having used the spectral theorem and the inverse Fourier formula for even functions on R. Applying the L 2 -norm on A t,j , we obtain ≤ h t − h t * g (2 j−1 √ t) −1 ∞ X β p t 1 B(2 j−1 √ t) L 2 (G) , by the spectral theorem. We estimate the supremum norm with the result of Lemma A.7: ∞ , and one checks easily For the L 2 -norm, the estimates in (A.2) for the heat kernel yields Hence we obtain M (1) t,j L 2 (A t,j ) We choose d to be the smallest positive integer such that d > a + n 2 + ln γ 0 2 so that we can sum up over j to obtain λ − m 2 + |f ( ) (λ)| < ∞. Then f 1 (λ) = (1 + λ) −N f (λ) satisfies the same properties as f but for m 1 = m − 2N and we can choose N large enough so that m 1 < 0. As f (λ) = f 1 (λ)(1 + λ) N , we also have f (λ π ) = f 1 (λ π )(1 + λ π ) N . If we knew that f 1 satisfies the property described in Proposition A.1 for m 1 and any q ∈ D(G) then this together with Lemmata 4.4 would imply the property for functions q such that ∆ q = ∆ [q] with a collection ∆ of order-1 difference operators satisfying the Leibniz-like property described in Definition 3.13. By Lemma 4.2 and Proposition 3.15, this would imply Proposition A.1 for f and any q ∈ D(G).
Reduction 2: we may assume f = 0 on [0, 2] as a consequence of the following property: Lemma A.8. For any q ∈ D(G) and m ∈ R, there exists C = C q,m > 0 and d = d [q],m ∈ N 0 such that for any function f ∈ C d [0, ∞) with support in [0, 2], we have ∀π ∈ G, t ∈ (0, 1) ∆ q f (tλ π ) L (Hπ) ≤ Ct Proof of Lemma A.8. From the properties of the Laplace operator and its Sobolev spaces together with (2.2), we have: (1 + λ π ) N ∆ q f (tλ π ) L (Hπ) = (1 + π(L)) N ∆ q f (tλ π ) L (Hπ) and, as q − b = m, the operators T j 's are uniformly bounded. We also need to find a bound for the operator norm of T j T * k whose convolution kernel is 2 (j+k) m 2 {L b 2 qg j (2 −j tL)δ 0 } * {L b 2 q * ḡ k (2 −k tL)δ 0 }. As the operator L is central, this kernel may be also written as 2 (j+k) m 2 {L b+c 2 qg j (2 −j tL)δ 0 } * {L b−c 2 q * ḡ k (2 −k tL)δ 0 } for any real number c. The estimate for L b 2 {qg j (2 −j tL)δ 0 } * in (A.15) holds in fact for any b ≥ 0 and by duality for any b ∈ R. Hence we can use it at b ± c to obtain